On asymptotics of Toeplitz determinants with symbols of nonstandard smoothness (Q1777950)

Let \(\mathbb T\) denote the unit circle. For a function \(a\in L^1(\mathbb T)\) and an integer \(n\geq 0\), the \(n\)-th Toeplitz determinant \(D_n(a)\) is given by \(D_n(a)=\det(a_{j-k})_{j,k=0}^n\) where \(a_i\) is the \(i\)-th Fourier coefficient of \(a\). In 1952, G. Szegő proved that if \(a\in C^{1+\epsilon}\) and \(a\geq 0\), then \[ \lim_{n\to \infty} {D_n(a) \over \exp[(n+1)(\log a)_0]}=\exp \left(\sum_{k=1}^\infty k(\log a)_k (\log a)_{-k}\right), \] which is nowadays called Szegő's strong limit theorem. The smoothness and positivity hypotheses needed by Szegő was then subsequently relaxed by many authors. The widest symbol class known so far, established by A. Böttcher and B. Silbermann in the early 1980's, is the subclass of the Wiener algebra \(W\) satisfying three conditions: (i) \(a(t)\neq 0\) for all \(t\in\mathbb T\), (ii) \(\text{Ind\,}a=0\), and (iii) \(a\in W\cap F\ell^{p, p/(p-1)}_{\alpha, 1-\alpha}\) with \(p\in(1,\infty)\) and \(\alpha\in [0,1]\), where \(F\ell^{p, p/(p-1)}_{\alpha, 1-\alpha}\) is the algebra consisting of all \(a\) such that \(\sum_{k=1}^\infty | a_{-k}(k+1)^\alpha| ^p+\sum_{k=0}^\infty | a_k(k+1)^{1-\alpha}| ^{p/(p-1)}<\infty\). In this paper, the authors extend the algebra \(W\cap F\ell^{p, p/(p-1)}_{\alpha, 1-\alpha}\) to the algebra \(W\cap F\ell^{\Phi, \Psi}_{\varphi, \psi}\) consisting of all \(a\in W\) such that \(\sum_{k=1}^\infty \Phi(| a_{-k}| \varphi_k)+\sum_{k=0}^\infty \Psi(| a_k| \psi_k)<\infty\), where \(\Phi\), \(\Psi\) are complementary \(N\)-functions satisfying a certain doubling condition and \(\varphi\), \(\psi\) are weight sequences satisfying some regularity and doubling conditions. They also construct an example showing that their algebra \(W\cap F\ell^{\Phi, \Psi}_{\varphi, \psi}\) strictly contains all \(W\cap F\ell^{p, p/(p-1)}_{\alpha, 1-\alpha}\).